اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدRandom matrix theory: Wigner-Dyson statistics and beyond. (Lecture notes of a course given at SISSA (Trieste, Italy))
168
0
0.0
(
0
)
تأليف
V.E.Kravtsov
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This is a course on Random Matrix Theory which includes traditional as well as advanced topics presented with an extensive use of classical logarithmic plasma analogy and that of the quantum systems of one-dimensional interacting fermions with inverse square interaction (Calogero-Sutherland model). Certain non-invariant random matrix ensembles are also considered with the emphasis on the eigenfunction statistics in them. The course can also be viewed as introduction to theory of localization where the (non-invariant) random matrix ensembles play a role of the toy models to illustrate functional methods based on super-vector/super-matrix representations.
قيم البحث
اقرأ أيضاً
We study the statistical properties of the complex generalization of Wigner time delay $tau_text{W}$ for sub-unitary wave chaotic scattering systems. We first demonstrate theoretically that the mean value of the $text{Re}[tau_text{W}]$ distribution f
unction for a system with uniform absorption strength $eta$ is equal to the fraction of scattering matrix poles with imaginary parts exceeding $eta$. The theory is tested experimentally with an ensemble of microwave graphs with either one or two scattering channels, and showing broken time-reversal invariance and variable uniform attenuation. The experimental results are in excellent agreement with the developed theory. The tails of the distributions of both real and imaginary time delay are measured and are also found to agree with theory. The results are applicable to any practical realization of a wave chaotic scattering system in the short-wavelength limit.
By calculating all terms of the high-density expansion of the euclidean random matrix theory (up to second-order in the inverse density) for the vibrational spectrum of a topologically disordered system we show that the low-frequency behavior of the
self energy is given by $Sigma(k,z)propto k^2z^{d/2}$ and not $Sigma(k,z)propto k^2z^{(d-2)/2}$, as claimed previously. This implies the presence of Rayleigh scattering and long-time tails of the velocity autocorrelation function of the analogous diffusion problem of the form $Z(t)propto t^{(d+2)/2}$.
We present a simple, perturbative approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit we call the Perturbative Resolvent Method (PRM). The PRM is based on constructing a linear system of equations and ca
lculating how the solutions to these equation change in response to a small perturbation using the zero-temperature cavity method. We illustrate the power of the method by providing simple analytic derivations of the Wigner Semi-circle Law for symmetric matrices, the Marchenko-Pastur Law for Wishart matrices, the spectral density for a product Wishart matrix composed of two square matrices, and the Circle and elliptic laws for real random matrices.
Theory of Random Matrix Ensembles have proven to be a useful tool in the study of the statistical distribution of energy or transmission levels of a wide variety of physical systems. We give an overview of certain q-generalizations of the Random Matr
ix Ensembles, which were first introduced in connection with the statistical description of disordered quantum conductors.
Although the spectra of random networks have been studied for a long time, the influence of network topology on the dense limit of network spectra remains poorly understood. By considering the configuration model of networks with four distinct degree
distributions, we show that the spectral density of the adjacency matrices of dense random networks is determined by the strength of the degree fluctuations. In particular, the eigenvalue distribution of dense networks with an exponential degree distribution is governed by a simple equation, from which we uncover a logarithmic singularity in the spectral density. We also derive a relation between the fourth moment of the eigenvalue distribution and the variance of the degree distribution, which leads to a sufficient condition for the breakdown of the Wigner semicircle law for dense random networks. Based on the same relation, we propose a classification scheme of the distinct universal behaviours of the spectral density in the dense limit. Our theoretical findings should lead to important insights on the mean-field behaviour of models defined on graphs.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
